MMN-3828

Bounds for the generalized elliptic integral of the second kind

  • Xiaohui Zhang, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China, xiaohui.zhang@zstu.edu.cn
  • Zhixia Xing, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China, xzxxiaoyuzhou@163.com

Abstract

For $a\in(0,1)$ and $r\in(0,1)$, let $\mathcal{E}_{a}(r)$ be the generalized elliptic integral of the second kind and $R(a)$ Ramanujan's constant. In this paper, we prove the following inequalities \begin{align*} \frac{\sin(\pi a)}{2(1-a)}+r'^2\left((1-a)\sin(\pi a)\log \left(\frac{e^{R(a)/2}}{r'}\right)-\gamma\right)<\mathcal{E}_{a}(r)\\ <\frac{\sin(\pi a)}{2(1-a)}+r'^2\left((1-a)\sin(\pi a)\log \left(\frac{e^{R(a)/2}}{r'}\right)-\delta\right) \end{align*} with the best possible constants $\gamma=\dfrac{1}{4}\sin(\pi a)$ and $\delta=\dfrac{\sin(\pi a)}{2(1-a)}+(1-a)\sin(\pi a)\dfrac{R(a)}{2}-\dfrac{\pi}{2}$.


Vol. 23 (2022), No. 1, pp. 495-503
DOI: 10.18514/MMN.2022.3828


Download: MMN-3828